User:Zeroparallax/Holor

A holor (Greek ὅλος "whole") is a mathematical entity that is made up of one or more independent quantities ("merates" as they are called in the theory of holors). Complex numbers, scalars, vectors, matrices, tensors, quaternions, and other hypercomplex numbers are kinds of holors. If proper index conventions are maintained then certain relations of holor algebra are consistent with that of real algebra; i.e. addition and uncontracted multiplication are both commutative and associative.

Overview
The term holor was coined by Parry Moon and Domina Eberle Spencer. Moon and Spencer classify holors as either nongeometric objects or geometric objects. They further classify the geometric objects as either oudors or akinetors, where the (contravariant) akinetors transform as


 * $$v^{i'} = \sigma {{\partial x^{i'}} \over {\partial x^{i}}} v^i,$$

and the oudors contain all other geometric objects (such as Christoffel symbols). The tensor is a special case of the akinetor where $$\sigma = 1$$. Akinetors correspond to pseudotensors in standard nomenclature.

Holors are furthermore classified with respect to their i) plethos n, and ii) valence N.

Moon and Spencer provide a novel classification of geometric figures in affine space with homogeneous coordinates. For example, a directed line segment that is free to slide along a given line is called a fixed rhabdor and corresponds to a sliding vector in standard nomenclature. Other objects in their classification scheme include free rhabdors, kineors, fixed strophors, free strophors, and helissors.